Pure rolling cycloids with variable effective diameter rollers

ABSTRACT

An apparatus includes a first ring having an open annular space and a variable-width groove disposed on an interior peripheral surface of the first ring; a second ring rotatable within the open annular space of the first ring, where the second ring has a respective variable-width groove disposed on an exterior peripheral surface of the second ring; and a plurality of rollers disposed between, and configured to roll on, the interior peripheral surface of the first ring and the exterior peripheral surface of the second ring and rotatable therebetween.

CROSS REFERENCE TO RELATED APPLICATION

The present application claims priority to U.S. Provisional patentapplication No. 62/185,502, filed on Jun. 26, 2015, and entitled “PureRolling Cycloids,” which is herein incorporated by reference as if fullyset forth in this description.

GOVERNMENT RIGHTS

This invention was made in part with U.S. Government support underContract No. W31P4Q-13-C-0046 awarded by the United States Army. TheGovernment may have certain rights in this invention.

BACKGROUND

The term “transmission” may refer generally to systems that providespeed and torque conversions from a rotating power source to anotherdevice. Industrial machinery, medical robotics, and domestic electronicsmay utilize such transmissions. Selecting or designing a transmissioninvolves considering multiple factors. Example factors include loadcapacity, efficiency, and cost.

SUMMARY

The present disclosure describes embodiments that relate to systems andapparatuses associated with pure rolling cycloids with variableeffective diameter rollers.

In one aspect, the present disclosure describes an apparatus. Theapparatus includes a first ring having an open annular space and aseries of variable-width cutouts disposed on an interior peripheralsurface of the first ring. The apparatus also includes a second ringrotatable within the open annular space of the first ring, where thesecond ring has a respective series of variable-width cutouts disposedon an exterior peripheral surface of the second ring. The apparatusfurther includes a plurality of rollers disposed between, and configuredto roll on, the interior peripheral surface of the first ring and theexterior peripheral surface of the second ring and rotatabletherebetween. The first ring has a total number of variable-widthcutouts and the second ring has a total number of variable-widthcutouts, with the total number of variable-width cutouts of the secondring being smaller than the total number of variable-width cutouts ofthe first ring, and a total number of the plurality of rollers beingless than the total number of variable-width cutouts of the first ringand greater than the total number of variable-width cutouts of thesecond ring.

In another aspect, the present disclosure describes an apparatus. Theapparatus includes a first ring having an open annular space and aplurality of depressions spatially arranged in series along an interiorperipheral surface of the first ring, where a groove is disposed in theinterior peripheral surface including the plurality of depressions. Theapparatus also includes a second ring rotatable within the open annularspace of the first ring, where the second ring has a respectiveplurality of depressions spatially arranged in series along an exteriorperipheral surface of the second ring, and where a respective groove isdisposed in the exterior peripheral surface including the respectiveplurality of depressions. The apparatus further includes a plurality ofrollers configured to engage, and roll within, the groove disposed inthe interior peripheral surface and the respective groove disposed inthe exterior peripheral surface as the second ring rotates within theopen annular space of the first ring.

In still another aspect, the present disclosure describes an apparatus.The apparatus includes a first ring having an open annular space and avariable-width groove disposed on an interior peripheral surface of thefirst ring. The apparatus also includes a second ring rotatable withinthe open annular space of the first ring, where the second ring has arespective variable-width groove disposed on an exterior peripheralsurface of the second ring. The apparatus further includes a pluralityof rollers disposed between, and configured to roll on, the interiorperipheral surface of the first ring and the exterior peripheral surfaceof the second ring and rotatable therebetween while engaging thevariable-width groove of the first ring and the respective variablewidth groove of the second ring.

The foregoing summary is illustrative only and is not intended to be inany way limiting. In addition to the illustrative aspects, embodiments,and features described above, further aspects, embodiments, and featureswill become apparent by reference to the figures and the followingdetailed description.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1A illustrates a circle rolling within a ring, in accordance withan example implementation.

FIG. 1B illustrates a cycloid curve resulting from rolling a circlewithin a ring, in accordance with an example implementation.

FIG. 2A illustrates a first ring, in accordance with an exampleimplementation.

FIG. 2B illustrates a second ring rotatable within an open annular spaceof the first ring shown in FIG. 2A, in accordance with an exampleimplementation.

FIG. 2C illustrates a cycloid drive apparatus with the second ring ofFIG. 2B rotatable within the first ring of FIG. 2A, in accordance withan example implementation.

FIG. 2D illustrates a simplified diagram of the apparatus shown in FIG.2C, in accordance with an example implementation.

FIG. 2E illustrates relationship between pitch circles and eccentricity,in accordance with an example implementation.

FIG. 2F illustrates similarity of triangles connecting instant center ofrotation with centers of pitch circles and contact points, in accordancewith an example implementation.

FIG. 3A illustrates a partial exploded view of the apparatus of FIG. 2Chaving non-spherical rollers, in accordance with an exampleimplementation.

FIG. 3B illustrates a cross section of a roller of the rollers shown inFIG. 3A, in accordance with an example implementation.

FIG. 4A illustrates an alternative second ring, in accordance with anexample implementation.

FIG. 4B illustrates a roller corresponding to the second ring of FIG.4A, in accordance with an example implementation.

FIG. 5A illustrates an alternative second ring, in accordance with anexample implementation

FIG. 5B illustrates a roller corresponding to the second ring of FIG.5A, in accordance with an example implementation.

FIG. 6A illustrates an alternative second ring having two side-by-sidegrooves, in accordance with an example implementation.

FIG. 6B illustrates a roller corresponding to the second ring of FIG.6A, in accordance with an example implementation.

FIG. 7A illustrates a knife-edge roller-groove configuration, inaccordance with an example implementation.

FIG. 7B illustrates a snug-fit roller-groove configuration, inaccordance with an example implementation.

FIG. 7C illustrates a chamfer roller-groove configuration, in accordancewith an example implementation.

FIG. 7D illustrates a gothic-arch roller-groove configuration, inaccordance with an example implementation.

FIG. 8 illustrates a perspective cross section of a differential cycloiddrive, in accordance with an example implementation.

FIG. 9A illustrates two cycloid drives connected in parallel, inaccordance with an example implementation.

FIG. 9B illustrates two cycloid drives connected in parallel and offsetrelative to each other, in accordance with an example implementation.

FIG. 10 illustrates an example cycloid drive configured to reducebacklash, in accordance with an example implementation.

FIG. 11 illustrates a coupling used to connect two shafts that are notaligned coaxially, in accordance with an example implementation.

FIG. 12A illustrates an exploded view of a coupling configuration tocompensate for eccentricity at an output of a cycloid drive, inaccordance with an example implementation.

FIG. 12B illustrates an exploded view of the coupling configuration ofFIG. 12A from another viewing angle, in accordance with an exampleimplementation.

FIG. 13A illustrates an exploded view of a configuration to compensatefor eccentricity at an output of a cycloid drive, in accordance with anexample implementation.

FIG. 13B illustrates an exploded view of the configuration of FIG. 13Afrom another viewing angle, in accordance with an exampleimplementation.

FIG. 14 illustrates another configuration to compensate for eccentricityof a cycloid drive, in accordance with an example implementation.

DETAILED DESCRIPTION

The following detailed description describes various features andfunctions of the disclosed systems and methods with reference to theaccompanying figures. The illustrative system and method embodimentsdescribed herein are not meant to be limiting. It may be readilyunderstood that certain aspects of the disclosed systems and methods canbe arranged and combined in a wide variety of different configurations,all of which are contemplated herein.

Further, unless context suggests otherwise, the features illustrated ineach of the figures may be used in combination with one another. Thus,the figures should be generally viewed as component aspects of one ormore overall implementations, with the understanding that not allillustrated features are necessary for each implementation.

Additionally, any enumeration of elements, blocks, or steps in thisspecification or the claims is for purposes of clarity. Thus, suchenumeration should not be interpreted to require or imply that theseelements, blocks, or steps adhere to a particular arrangement or arecarried out in a particular order.

By the term “substantially” it is meant that the recited characteristic,parameter, or value need not be achieved exactly, but that deviations orvariations, including for example, tolerances, measurement error,measurement accuracy limitations and other factors known to skill in theart, may occur in amounts that do not preclude the effect thecharacteristic was intended to provide.

I. Overview

Selecting or designing a transmission for a particular applicationinvolves considering multiple factors. Example factors include loadcapacity, efficiency, and cost. Transmission systems could be heavy ifdesigned for a large load capacity. On the other hand, smalltransmission systems tend to have a small load capacity. Further,transmission systems tend to be expensive if high performance, definedby parameters such as efficiency, backlash, etc., is desired. Hence, atransmission system that can increase load capacity for a given size andalso reduce the cost of manufacturing can be beneficial.

In some examples, transmissions are designed to meet some goals at theexpense of others. For instance, ball or roller bearings could be usedin transmissions systems to achieve high efficiency. However, thesebearings experience stress concentrations due to the small sizes oftheir rollers. To alleviate stress concentrations, fixed slidingcontacts could be used instead, but sliding contacts typically havelower efficiency.

Disclosed herein are transmissions involving cycloid drive apparatusesand systems that utilize pure rolling components that could be largerthan roller bearings, thus providing large torque capacity for a givenweight of the transmission. Further, the disclosed systems utilizeconfigurations to harvest the output of the cycloid drives andcompensate for inherent eccentricity of cycloid drives.

II. Cycloid Motion

As used herein, the term “cycloid” refers to the curve traced by a pointon a rim of a circular wheel as the wheel rolls along either a straightor circular path without slippage. In an example, cycloid motion resultswhen the circular wheel rolls inside a main circle or ring. FIG. 1Aillustrates a circle 100 rolling within a ring 102, in accordance withan example implementation. A point 104 on a rim of the circle 100 tracesa cycloid curve as the circle 100 moves along an internal surface of thering 102.

FIG. 1B illustrates a cycloid curve 106 resulting from rolling thecircle 100 within the ring 102, in accordance with an exampleimplementation. The cycloid curve 106 is traced by the point 104 as thecircle 100 rolls within the ring 102. The motion of the circle 100,while the ring 102 remains stationary, could be referred to as cycloidmotion.

Disclosed herein are example transmission systems and apparatuses thatutilize this cycloid motion. These systems and apparatuses can providean advantageous configuration that may achieve high efficiency and lightweight or small form factor. These transmission systems could be used inrobotic applications where motors and transmissions could be mounted ata distance from the main body of a robot. Automotive, heavy industry,and energy generation, among other applications, could also benefit fromutilizing the transmissions described herein.

III. Example Cycloid Drive Apparatus

FIGS. 2A, 2B, 2C, 2D, 2E, and 2F illustrate an example cycloid driveapparatus, in accordance with an example implementation. Particularly,FIG. 2A illustrates a first ring 200, in accordance with an exampleimplementation. The first ring 200 has an open annular space 202 and aseries of variable-width cutouts 204A, 204B, 204C, 204D, 204E, and 204Fdisposed on an interior peripheral surface 206 of the first ring 200.

FIG. 2B illustrates a second ring 208 that is rotatable within the openannular space 202 of the first ring 200, in accordance with an exampleimplementation. The second ring 208 has a respective series ofvariable-width cutouts such as cutout 210A, 210B, 210C, and 210Ddisposed on an exterior peripheral surface 212 of the second ring 208.

Each cutout of the series of variable-width cutouts 204A-F of the firstring 200 and the series of variable-width cutouts 210A-D of the secondring 208 starts with a first width at a first end of the cutout. Thewidth then increases to a second width larger than the first width at acenter of the cutout, and then narrows back to the first width at asecond end of the cutout. To illustrate, the cutout 210A of the secondring 208 has a first end 214A and a second end 214B. The width of thecutout 210A at the first end 214A is small. The width then increasesgradually to a width “d” at a center of the cutout 210A, then decreasesgradually until the second end 214B, where the width is similar to thewidth at the first end 214A.

FIGS. 2A and 2B illustrate separate cutouts 204A-F and 210A-D that areseparated by blank areas of the interior peripheral surface 206 and theexterior peripheral surface 212, respectively. For instance, referringto the first ring 200 illustrated in FIG. 2A, the cutouts 204A-F aredistinct and separate from each other and are separated by blank regionson the interior peripheral surface 206.

However, in other example implementations the surfaces 206 and 212 mayeach have a respective continuous variable-width channel or groovedisposed therein. Each continuous variable-width channel or groove maybe analogous to a raceway of a bearing. In this analogy, the rings 200and 208 operate similar to races of the bearing. A width of thevariable-width groove may vary gradually between a first width and asecond width larger than the first width. For instance, the first widthmay be similar to the width at the first end 214A of the cutout 210A,and the second width may be similar to the width “d” at the center ofthe cutout 210A. The variable-width cutouts 204A-F and 210A-D mayrepresent regions of the variable-width groove that increase from thefirst width to the second width and back to the first width. Thevariable-width cutouts 204A-F and 210A-D may then be separated byportions such as a portion of the variable-width groove having the firstwidth or some other width (see e.g., FIG. 5A). In this manner, thevariable-width cutouts 204A-F and 210A-D may be portions of respectivevariable-width continuous grooves or raceways.

FIG. 2C illustrates a cycloid drive apparatus 216 with the second ring208 rotatable within the first ring 200, in accordance with an exampleimplementation. The apparatus 216 includes a roller cage 218 that isdisposed between the first ring 200 and the second ring 208 andconfigured to couple a plurality of rollers 220A, 220B, 220C, 220D, and220E to each other. The roller cage 218 is rotatable in the open annularspace 202 of the first ring 200 as the plurality of rollers 220A-E rollon and between the interior peripheral surface 206 of the first ring 200and the exterior peripheral surface 212 of the second ring 208. Theroller cage 218 couples the rollers 220A-E such that the rollers 220A-Eare equidistant from each other.

As each roller of the rollers 220A-E roll on the interior peripheralsurface 206 and the exterior peripheral surface 212 of the second ring208, the roller traverses the cutouts 204A-F and 210A-D. As a roller ofthe rollers 220A-E traverse a cutout of the cutouts 204A-F and 210A-D,the roller moves from an area of the cutout that has a small width to anarea that is wider (i.e., near a center region of the cutout). Thus, asthe roller passes over the cutout, more or less of the roller engages inthe cutout. Particularly, at the wider area of the cutout, the rollerpasses deeper through the surface that the cutout is disposed in, i.e.,the interior peripheral surface 206 or the exterior peripheral surface212. Thus, as the roller traverses through the cutout, a radial distancebetween a center of the first ring 200 and the roller varies.

To illustrate, as shown in FIG. 2C, the roller 220E is less engaged withthe cutout 204B as the roller 220E is near one end of the cutout 204B.On the other hand, the roller 220D is more engaged with the cutout 204C,i.e., the roller 220D is disposed deeper in the cutout 204C as theroller 220D nears the center of the cutout 204C. The roller 220C is evenmore engaged with the cutout 204D than the roller 220D is engaged withthe cutout 204C as the roller 220C is substantially at the center of thecutout 204D. The rollers 220A-E behave similarly and engage more or lesswith the cutouts 210A-D of the second ring 208 as the rollers 220A-Etraverse the cutouts 210A-D. The variable width of the cutouts orgrooves defining the cutout is the means by which the variable effectivediameter of the rollers is generated as the rollers traverse the cutoutsor grooves. These variable effective diameter rollers enable maintainingpure rolling motion in the apparatus 216.

As mentioned above, the cutouts 204A-F and 210A-D could be parts of arespective variable-width groove that operates analogously to theraceways of a bearing. Thus, in principle, each of the rollers 220A-Econtacts each raceway at a single point. However, a load on aninfinitely small point would cause infinitely high contact pressure. Inpractice, the roller deforms (flattens) slightly where it contacts eachraceway, much as a tire flattens where it touches the road. The racewayalso dents slightly where each roller presses on it. Thus, the contactbetween roller and raceway is of finite size and has finite pressure.

In an example, the apparatus 216 could operate as a cycloid speedreducer configured to reduce the speed of an input shaft by a certainratio. For instance, the second ring 208 could be eccentrically mountedvia a bearing to an input shaft (not shown). In this configuration, theinput shaft drives the second ring 208 along a curved path within theopen annular space 202 of the first ring 200. Further, in an example,the first ring 200 could be configured as a stator of the cycloid speedreducer (i.e., the first ring 200 could be fixed). Then, an output shaftcould be coupled to the second ring 208, with the output shaft having areduced speed compared to the input shaft. In another example, the firstring 200 could be coupled to an output shaft and rotatable, whereas thesecond ring 208 could be fixed and configured to operate as the statorof the cycloid speed reducer. Thus, the input, output, and statordesignations are interchangeable.

For the apparatus 216 to operate as a cycloid speed reducer, the totalnumber of variable-width cutouts 210A-D of the second ring 208 is lessthan the total number of variable-width cutouts 204A-F of the first ring200. Further, a total number of the rollers 220A-E is less than thetotal number of variable-width cutouts 204A-F of the first ring 200 andgreater than the total number of variable-width cutouts 210A-D of thesecond ring 208. In the apparatus 216 described above, the first ring200 has six cutouts, the second ring 208 has four cutouts, and fiverollers 220A-E are disposed between the first ring 200 and the secondring 208.

The reduction ratio of the cycloid speed reducer is determined based onthe total number of rollers 220A-E. Particularly, the reduction ratiocould be calculated using the following equation:

$\begin{matrix}{R = \frac{N_{r} - 1}{2}} & (1)\end{matrix}$where R is the reduction ratio and N_(r) is the number of rollers.

One advantage of the apparatus 216 is that, based on equation (1), theapparatus 216 is capable of providing non-integer reduction ratios. Asan example, if the first ring 200 has seven cutouts, the second ring 208has five cutouts, and six rollers are disposed between the first ring200 and the second ring 208 the ratio R can be calculated by equation(1) to be 2.5:1.

In examples, the total number of cutouts and rollers are threeconsecutive integers, e.g. 4 4 cutouts for the second ring 208, 5rollers in the roller cage 218, and 6 cutouts in the first ring 200, asillustrated in FIGS. 2A-2C. However, cycloid drives with other patternse.g. 4 cutouts for the second ring 208, 6 rollers in the roller cage218, and 8 cutouts in the first ring 200, are also possible.

IV. Rolling Behavior

Pure rolling occurs when both the magnitude and direction of the linearvelocities of rigid bodies at their contact points match. FIG. 2Dillustrates a simplified diagram of the apparatus 216, in accordancewith an example implementation. The following analysis holds for anynumber of rollers or reduction ratio. FIG. 2D facilitates analysis ofthe apparatus 216 and relation between parameters that achieve purerolling of the rollers 220A-E.

FIG. 2D illustrates the first ring 200, the second ring 208, and theroller cage 218 as circles or cylinders. The cylinders of the first ring200 and the second ring 208 define the surfaces that the roller 220A-Emake contact with. Thus, these cylinders will lie within the grooves orraceways of the first ring 200 and the second ring 208. Each of thefirst ring 200, the second ring 208, and the roller cage 218 has acorresponding pitch circle that is rigidly and concentricallyrespectively attached thereto. In FIG. 2D, pitch circle 222 correspondsto the first ring 200, pitch circle 224 corresponds to the second ring208, and pitch circle 226 corresponds to the roller cage 218.

The surfaces that define the rings 200 and 208 are described here ascylinders, but they can also be conic sections, where each cross sectionis a circle. This implementation would be analogous to bevel gear typearrangements.

The three pitch circles 222, 224, and 226 could be defined by a desiredreduction ratio to be achieved by the apparatus 216 and an amount ofeccentricity between the input shaft and the second ring 208.Specifically, the ratio of the pitch circle diameters and the ratio ofthe diameters of the first ring 200, the second ring 208, and the rollercage 218 are equal to the ratio of the integer number of cutouts orrollers that each component has. For example, the ratio between thediameter of the pitch circle 222 and the diameter of the pitch circle224 is equal to a ratio between the number of cutouts in the first ring200 and the number of cutouts in the second ring 208. Similarly, theratio between the diameters of the first ring 200 and the second ring208 is also equal to the ratio between the number of cutouts in thefirst ring 200 and the number of cutouts in the second ring 208. Asanother example, the ratio between the diameter of the pitch circle 222and the diameter of the pitch circle 226 is equal to the ratio betweenthe number of cutouts in the first ring 200 and the number of rollerscoupled to the roller cage 218. Similarly, the ratio between thediameters of the first ring 200 and the roller 218 is also equal to theratio between the number of cutouts in the first ring 200 and the numberof rollers coupled to the roller cage 218.

Another constraint that facilitates defining the pitch circles 222, 224,226 is the amount of eccentricity between the input shaft and the secondring 208. FIG. 2E illustrates relationship between the pitch circles222, 224, and 226 and the amount of eccentricity, in accordance with anexample implementation. Specifically, a difference 227A between theradius of the pitch circle 222 and the radius of the pitch circle 226 ishalf the amount of eccentricity. Further, a difference 227B between theradius of the pitch circle 222 and the radius of the pitch circle 224 isequal to the amount of eccentricity. As an example for illustration, theamount of eccentricity for the apparatus 216 with the above-mentionednumber of cutouts and rollers could be 20 millimeter. In this example,the radius of the pitch circle 222 could be 60 mm, the radius of thepitch circle 226 could be 50 mm, and the radius of the pitch circle 224could be 40 mm.

Pitch circles that are defined based on both the desired reduction ratioto be achieved by the apparatus 216 and the amount of eccentricity asdiscussed above, would intersect at a common instant center of rotation228. The instant center of rotation 228, which could also be referred toas the instantaneous velocity center, is the point fixed to a bodyundergoing planar movement, with the point having zero velocity at aparticular instant of time. At this instant, the velocity vectors of thetrajectories of other points in the body generate a circular fieldaround this point which is identical to what is generated by a purerotation about the point.

As the first ring 200, the second ring 208, and the roller cage 218share the instant center of rotation 228, contact points of a rollerwith the first ring 200 and the second ring 208 exist along a line thatpasses through the center of the roller, and also through the instantcenter of rotation 228. For example, that line for the roller 220D isrepresented by line 230 in FIG. 2E.

As a result of this configuration the direction of the velocities at thecontact points between rigid bodies in the apparatus 216 (e.g., betweenthe roller 220D, the first ring 200 and the second ring 208) match. Inother words, the rigid bodies will not intersect or disconnect at thefollowing moment. This can be referred to as the non-interferencecondition for pure rolling. However, at least one other condition is metfor pure rolling to occur. Particularly, for pure rolling to occur, inaddition to matching the direction of the velocities, absolutevelocities (i.e., scalar magnitude of velocities) of the rigid bodypairs at their contact points should also match. Each contact point isanother constraint on the motion of the large set of rigid bodiesconnected at that point.

The absolute velocity of any point on a rigid body can be found bymultiplying the angular velocity of the rigid body by the distancebetween the point and the instant center of rotation for the rigid body.Therefore, for the absolute velocities to match given a known set ofangular velocities of the rigid bodies, these contact points should belocated at specific distances from the instant center of rotation so asto preclude slipping.

For the apparatus 216, each of the rollers 220A-E contacts both thefirst ring 200 and the second ring 208 as it rolls therebetween. Theresulting two contact points between a roller and the first and secondrings 200 and 208 respectively should have the same angular velocity inorder to preclude slipping.

Generally, the instant center of rotation for an object is dependent onits fixed frame of reference. Because all three pitch circles 222, 224,and 226 share a common instant center of rotation, any pitch circle canbe chosen as the frame of reference. Choosing the pitch circle 226 thatrepresents the roller cage 218 as a fixed frame of reference has anadded advantage for the analysis presented herein. Treating the pitchcircle 226 as a fixed frame of reference indicates that a respectivecenter of each roller is now fixed, and its instant center of rotationis coincident with its geometric center. For instance, geometric center231 of the roller 220D coincides with its instant center of rotation.However, the results would be the same for other configurations wherefor example the pitch circle 222 is fixed.

Thus, for the purposes of this analysis, it is convenient to fix theroller cage 218 and allow the first and second rings 200 and 208 torotate. As shown in FIG. 2E, the two contact points 232A and 232B are atequal distances from the instant center of rotation, which is coincidentwith the geometric center 231, of the roller 220D. Therefore, referringto the roller 220D in FIG. 2E, the absolute velocities at two contactpoints 232A and 232B should be equal and opposite to preclude the roller220D from slipping.

Centers of the three pitch circles 222, 224, and 226 are at a fixeddistance away from each other. Thus, the distance between centers 234Aand 234B of the pitch circles 222 and 224 is equal to the amount ofeccentricity (i.e., 227B), and they are at equal distances from thecenter of the fixed frame of reference, i.e., center 235 the pitchcircle 226. To preclude slipping, the absolute velocities of the centers234B and 234A should therefore be equal. Because the centers 234A and234B are the centers of the pitch circles 222 and 224, another equationcan be written to describe their velocities:B*w ₃ =A*w ₃  (2)where “B” is a radius of the pitch circle 222 and “A” is a radius of thepitch circle 224, “w₁” is the angular velocity of the pitch circle 222and the first ring 200, and “w₃” is the angular velocity of the pitchcircle 224 and the second ring 208. The angular velocities “w₁” and “w₃”can be determined from the desired reduction ratio, but as shown in theanalysis below, these angular velocities drop out of this analysis andmight not be determined to show pure rolling.

As illustrated in FIG. 2E, “B” and “A” are collinear, i.e., a line 233extending from the instant center of rotation 228 and intersecting withthe centers 234A and 234B of the pitch circles 222 and 224,respectively, exists. Based on equation (2):

$\begin{matrix}{\left. \Rightarrow\frac{B}{A} \right. = \frac{w_{1}}{w_{3}}} & (3)\end{matrix}$

FIG. 2F illustrates similarity of triangles 236A and 236B connecting theinstant center of rotation 228 with the centers 234A and 234B of thepitch circles 222 and 224 and the contact points 232A and 232B, inaccordance with an example implementation. The triangles 236A and 236Bshare the angles θ and ϕ as shown in FIG. 2F. Based on the illustratedsimilarity of the triangles 236A and 236B:

$\begin{matrix}{\left. \Rightarrow\frac{D}{C} \right. = \frac{B}{A}} & (4)\end{matrix}$where “D” is a distance between the contact point 232A and the instantcenter of rotation 228 and “C” is a distance between the contact point232B and the instant center of rotation 228. Thus:

$\begin{matrix}{{\left. \Rightarrow\frac{D}{C} \right. = \frac{w_{1}}{w_{3}}}{{or}\text{:}}} & (5) \\{{D*w_{3}} = {C*w_{1}}} & (6)\end{matrix}$

Referring back to FIG. 2D:D*w ₃ =V ₁ and C*w ₁ =V ₃  (7)where “V₃” and “V₁” are linear scalar velocities at the contact points232A and 232B, respectively. Based on equations (6) and (7):V ₁ =V ₃  (8)

According to equation (8) the velocities “V₃” and “V₁” at the contactpoints 232A and 232B, respectively, are equal in magnitude, andtherefore are consistent with the constraints provided by the roller20D, and thus no slipping would occur.

FIGS. 2D and 2E represent a snapshot of a dynamic geometricconfiguration, where the instant center of rotation 228 completes anorbit for each rotation of the input shaft coupled to the cycloid speedreducer. The effective diameters of the rollers 220A-E are constantlychanging through an orbit. Nonetheless, the above analysis relies oninvariant parameters, and thus holds for all configurations through acycle, and therefore all roller contact points.

Thus, the rollers 220A-E of the apparatus 216 should roll withoutslipping if the apparatus 216 has the dimensional relationshipsdescribed above, e.g., by equations (2)-(8), and illustrated in FIGS.2D, 2E, and 2F.

Referring to FIGS. 2D-2F, there appears to be interference between therollers 220A-E and the first ring 200 and the second ring 208. Forinstance, a region 238 shown in FIG. 2F appears to be an interferencebetween the roller 220B and the first ring 200. However, the region 238is not an interference. The region 238 illustrates that the roller 220Bsits deeper in a respective groove or cutout (e.g., a cutout of thecutouts 204A-F) in the first ring 200. In other words, the roller 220Bhappens to be at a point in the groove or cutout that is sufficientlywide, causing the roller 220B to sink deeper in the first ring 200.

V. Example Alternative Configurations for the Cycloid Drive Apparatus

Several example alternative configurations for the apparatus 216described in FIGS. 2A-2F are now described. In the configurationdescribed above, the rollers 220A-E are shown to be spherical. FIGS.3A-3B illustrate use of non-spherical rollers, in accordance with anexample implementation.

Particularly, FIG. 3A illustrates a partial exploded view of theapparatus 216 showing non-spherical rollers 300, in accordance with anexample implementation. The rollers 300 replace the rollers 220A-E andare configured to roll between the first ring 200 and the second ring208.

FIG. 3B illustrates a cross section of a roller of the rollers 300, inaccordance with an example implementation. As shown, the cross sectionA-A′ is rhombic in a plane parallel to an axis of rotation of the secondring 208, i.e., in a plane parallel to an arrow 302. The cross sectionin a plane perpendicular to the axis of rotation of the second ring 208,i.e., perpendicular to the arrow 302, is a circular cross sectionsimilar to cross sections of the rollers 220A-E. The grooves or cutoutsof the first ring 200 and the second ring 208 could also be changed tomatch a shape of the rollers 300. Other shapes of the rollers andgrooves are also possible. However, one constraint on the shape of aroller is that the roller should be symmetric about its axis ofrotation.

FIG. 4A illustrates an alternative second ring 400, and FIG. 4Billustrates a corresponding roller 402, in accordance with an exampleimplementation. Instead of having variable-width cutouts like the secondring 208, the second ring 400 has a plurality of depressions such asdepressions 404 spatially arranged in series along an exteriorperipheral surface 406 of the second ring 400. A groove or channel 408is disposed along the exterior peripheral surface 406. The channel 408dips along with the depressions 404 as shown in FIG. 4A. A correspondingfirst ring (not shown) would have similar depressions and channel tomatch the second ring 400 and the roller 402.

FIG. 5A illustrates an alternative second ring 500, and FIG. 5Billustrates a corresponding roller 502, in accordance with an exampleimplementation. The second ring 500 has a variable-width groove 504disposed on an exterior peripheral surface 506 of the second ring 500.Width of the variable-width groove 504 varies in a spatially periodicmanner between a first width “d₁” and a second width “d₂” larger thanthe first width “d₁”. A corresponding first ring (not shown) would havea similar groove to match the second ring 500 and the roller 502.

The roller 502 has a shape that matches a profile of the variable-widthgroove 504. Similar to the rollers 300, the roller 502 may have arhombic cross section in a plane parallel to an axis of rotation of thesecond ring 500. However, the roller 502 has longer conical portions508A-508B compared to the rollers 300. The longer conical portions508A-508B may impart more consistent stiffness across the roller 502 asit rolls along the variable-width groove 504.

FIG. 6A illustrates an alternative second ring 600 having twoside-by-side grooves 602A and 602B, and FIG. 6B illustrates acorresponding roller 604, in accordance with an example implementation.The grooves 602A and 602B are disposed on an exterior peripheral surface606 of the second ring 600. Like the groove 504, respective widths ofthe grooves 602A and 602B vary in a spatially periodic manner between afirst width “d₃” and a second width “d₄” larger than the first width“d₃”. A corresponding first ring (not shown) would have similar parallelgrooves to match the grooves 602A and 602B of the second ring 600 andthe roller 602.

The roller 604 has a shape that matches respective profiles of thegrooves 602A and 602B. Particularly, the roller 604 may be composed oftwo rollers 608A and 608B disposed side-by-side. In examples, therollers 608A and 608B may be coupled to each other; however, in otherexamples, they might not be coupled to each other.

In examples, more than two grooves and more than two rollers could bestacked side-by-side. This construction of the roller 604 increases loadcapacity of the cycloid drive because the load is distributed among agreater number of contact points.

Further, using the side-by side rollers 608A-608B facilitates reducingtheir diameters for a given load capacity. As a result of using smallerroller diameters, the eccentricity of the input shaft relative to thesecond ring 600 can be decreased. Although the eccentricity can beremoved at the output stage (see FIGS. 11-14), smaller eccentricity maybe easier to reduce or remove, and the cycloid drive might be subjectedto less vibration.

The implementations described in FIGS. 2A-6B are examples forillustration and other example implementations are contemplated. Forexample, it is contemplated that an apparatus may have the second ring208 without the first ring 200. In this example, the second ring 208 mayhave a variable-width groove and the rollers 220A-E may traverse thevariable-width groove, thus changing an effective diameter of therollers 220A-E. In other examples, the rings 200 and 208 might not becircular in shape, but may have other non-circular shapes. For instance,the surfaces that define the rings 200 and 208 could be conic sections.This implementation would be analogous to bevel gear type arrangements.In another example, the variable-width grooves may trace a helical pathinstead of a circular path about a peripheral surface of a ring. In thisexample, the rollers would follow the helical path instead of a circularpath shown in the Figures discussed above. Other implementations arepossible as well.

VI. Example Roller-Groove Configurations

The configurations described above with respect to FIGS. 2A-2C, 3A-3B,4A-4B, 5A-5B, and 6A-6B illustrate various example implementations of acycloid drive with variable effective diameter rollers. Various othertypes of grooves, cutout, or channels could be used along withcorresponding roller configuration and shapes. Further, other rollercross sections that maintain axial symmetry of the roller could be used.

The roller configuration, shape, and profile and the correspondingconfiguration of the grooves impact load capacity, load sharing,stiffness, efficiency and friction, contact stresses, torque output ofthe cycloid drive, and kinematic constraints of the cycloid drive. Thus,the configuration of the roller may be considered as a design parameterthat may be adjusted to balance various requirements, such as loadcapacity and efficiency of the cycloid drive.

FIGS. 7A-7D illustrate example roller-groove configurations. The term“groove” is used in this section with respect to FIGS. 7A-7D toencompass the terms “cutouts,” “channels,” and “raceways” in addition togrooves. As mentioned above, the grooves are analogous to raceways ofbearings, and the rings are analogous to races of bearings.

FIGS. 7A-7D illustrate cross section views of a roller 700 restingbetween an outer groove 702 of an outer or first ring 704 and an innergroove 706 of an inner or second ring 708. FIGS. 7A-7D illustrate fourdifferent roller-groove example designs. Although the roller 700 isshown as a spherical roller, other roller shapes could be used. Theroller 700 may represent any of the rollers discussed above. Also, thefirst ring 704 may represent any of the first rings described above andthe second ring 708 may represent any of the second rings discussedabove.

The roller 700 interfaces with the grooves 702 and 706 at four locations710A, 710B, 710C, and 710D. This interface can be optimized for a numberof qualities such as efficiency, load capacity, wear, etc.

FIG. 7A illustrates a knife-edge roller-groove configuration, inaccordance with an example implementation. The knife-edge configurationshown in FIG. 7A is characterized by the interface locations 710A-710Dbeing substantially single-point contact or contact patches that spreadto a short line contact under high loads. This style could provide highefficiency at all loads, but may be associated with increased wear.

FIG. 7B illustrates a pocket or snug-fit roller-groove configuration, inaccordance with an example implementation. In this snug-fitconfiguration, each of the locations 710A-710D has a longer line contactpatch with length “w.” This configuration can have high load capacity.At high loads, the line contact patches at the interface locations710A-710D gets slightly thicker, thus increasing load capacity. However,this interface configuration causes lower efficiency due to slightscrubbing between the roller 700 and the rings 704 and 708.

FIG. 7C illustrates a chamfer roller-groove configuration, in accordancewith an example implementation. This interface starts out as a pointcontact under low pressure and grows to an ellipsoidal shape under load.This interface may be easier to manufacture for some roller shapes dueto simplified geometry.

FIG. 7D illustrates a gothic-arch roller-groove configuration, inaccordance with an example implementation. In the configuration shown inFIG. 7D, the grooves 702 and 706 may have a curvature that is less thanthat of the roller 700 such that the contact patches at the locations710A-710D grows quicker under load while maintaining high efficiency.

In examples, the configurations shown in FIG. 7A-7D could be usedindividually and consistently within a cycloid drive; however, theseconfigurations may be combined within the cycloid drive for optimaleffect along the length of a groove (i.e., the groove 702 and/or thegroove 706). For instance, the widest part of the groove may have achamfer interface configuration shown in FIG. 7C, while the highest loadcapacity part of the groove could have the snug-fit configuration shownin FIG. 7B. In examples, relief could be added to sections of the grooveso as to temporarily loosen contact with the roller 700. This could bedone to allow for re-alignment of parts, or reduction in wear.

VII. Example Parallel and Differential Cycloid Drive Configurations

FIG. 8 illustrates a perspective cross section of a differential cycloiddrive 800, in accordance with an example implementation. Thedifferential cycloid drive 800 includes two cycloid drives 802 and 804connected differentially as shown in FIG. 8. The cycloid drives 802 and804 could have any of the configurations discussed above and havedifferent reduction ratios relative to each other.

A second ring 806 of the first cycloid drive 802 is coupled or connectedto a second ring 808 of the second cycloid drive 804 via a couplingmember 810. An input shaft (not shown) is configured to be eccentricallycoupled to bearings or drive members 812A and 812B. The input shaft andthe drive member 812A drive the second ring 806 within a fixed orstationary first ring 814 of the first cycloid drive 802. In otherwords, the first ring 814 is considered mechanical ground for thedifferential cycloid drive 800. In contrast, a first ring 816 of thesecond cycloid drive 804 is free to rotate.

This configuration allows for large reduction ratios. Specifically,assuming the first cycloid drive 802 has a reduction ratio “R₁,” and thesecond cycloid drive 804 has a reduction ratio “R₂,” the resultingreduction ratio R_(d) of the differential cycloid drive 800 can bedetermined by the following equation:

$\begin{matrix}{R_{d} = \frac{R_{1} - R_{2}}{1 + R_{2}}} & (9)\end{matrix}$For instance, if “R₁” is 2:1 (i.e., R₁=½) and “R₂” is 2.5:1

$\left( {{i.e.},{R_{2} = \frac{1}{2.5}}} \right),$then R_(d) can be calculated by equation (9) to be 1/14. As such, largereduction ratios are achievable with the differential cycloid drive 800.

Another advantage of the differential cycloid drive 800 is that theeccentricity of the first cycloid drive 802 is cancelled or compensatedfor by the respective eccentricity of the second cycloid drive 804. Inthis manner, no additional mechanism are coupled to the differentialcycloid drive 800 to rectify the output (i.e., compensate for theeccentricity) at the first ring 816.

Differential cycloids such as the differential cycloid drive 800 includethe two cycloid drives 802 and 804 connected differentially. The cycloiddrives 802 and 804 could also be connected in parallel or stackedtogether. FIG. 9A illustrates the two cycloid drives 802 and 804connected in parallel, in accordance with an example implementation. Theconfiguration shown in FIG. 9A is similar to the configuration shown inFIGS. 6A-6B. The drive members 812A and 812B are eccentrically mountedto an input shaft 902 and are configured to drive their respectivesecond rings 806 and 808.

Using multiple pure rolling cycloids like the cycloid drives 802 and 804stacked on top of each other in the same orientation, i.e., stacked inparallel, allows for greater load capacity as the load is distributedamong multiple cycloid drives without significantly increasing partcount. In an example, to facilitate manufacturing of this configuration,matched components from each cycloid could be manufactured as a singlepart. For instance, a single external ring could be manufactured toreplace the two first rings 814 and 816. Similarly, a single internalring, such as the ring 600 illustrated in FIG. 6A, could be manufacturedto replace the two second rings 806 and 808.

In examples, multiple pure rolling cycloids could be stacked on top ofeach other while being offset relative to each other. FIG. 9Billustrates the two cycloid drives 802 and 804 connected in parallelwith the cycloid drive 802 being offset relative to the cycloid drive804, in accordance with an example implementation. As shown in FIG. 9B,the drive member 812A and the second ring 806 of the cycloid drive 802are shifted upward relative to the drive member 812B and the second ring808 of the cycloid drive 804. This configuration allows for greater loadcapacity and more even load transfer. However, the outputs connected tothe second rings 806 and 808 might not be joined into a single output,and might be harvested independently.

VIII. Example Implementation with Backlash Reduction

It is desirable to reduce or eliminate backlash in mechanical systems.One source of backlash in the apparatuses and systems described above isthe manufacturing tolerance between the rollers and the channels,grooves, or cutouts.

FIG. 10 illustrates an example cycloid drive 1000 configured to reducebacklash, in accordance with an example implementation. A first (outer)ring 1002 of the cycloid drive 1000 may be split into two halves 1004Aand 1004B. The two halves 1004A-1004B could be coupled together by stiffsprings or screws such as screw 1006. By tightening the screw(s) 1006, awidth of grooves (channels or cutouts) 1008 and 1010 may be reduced,thus causing interference between the grooves 1008, 1010 and roller(s)1012. The tighter the screw(s), the more reduction in backlash isachieved. However, efficiency is reduced because friction increasesbetween the roller(s) 1012 and the grooves 1008 and 1010.

The cycloid drive 1000 could also be used as an integrated overridingclutch. When the input torque exceeds a threshold, the output slipsrelative to the input. To use the cycloid drive 1000 as an integratedoverriding clutch, the screws 1006 connecting the two halves 1004A-1004Bcould be replaced with stiff springs that can displace significantly.When the input torque exceeds a threshold value, the two halves 1004Aand 1004B may move apart from each other, allowing the roller(s) 1012 topass through a groove (e.g., the grooves 1008 and 1010) that wouldotherwise be too small for the roller(s) 1012 to pass therethrough. Asthe roller(s) 1012 traverse a groove, they may be caught in the nextsection of the groove, unless the input torque still exceeds thethreshold value. An advantage of this clutch is that it could beintegrated directly in to the transmission with few added parts.

The cycloid drive 1000 could be configured by splitting the outer ring1002 as shown in FIG. 10, inner ring 1014, or both.

IX. Compensating for Eccentricity

As mentioned above, the second (internal) ring (e.g., any of the secondrings 208, 400, 500, 600, and 806) of cycloid drives are mountedeccentrically to an input shaft via a bearing or drive member. An outputshaft of the cycloid drive is coupled to the second ring of the cycloiddrive, and therefore the output shaft has eccentric motion and displacesin a perpendicular direction to the rotational axis of the second ring.In order to transmit a concentric angular rotation, the displacement canbe removed.

FIG. 11 illustrates a coupling 1100 used to connect two shafts that arenot aligned coaxially, in accordance with an example embodiment. Thecoupling 1100 includes three disks 1102, 1104, and 1106. One of theouter disks such as the disk 1102 may be coupled to an input shaft whilethe other outer disk 1106 may be coupled to an output shaft.

The middle disk 1104 is coupled to both outer disks by tongue (i.e.,key) and groove (i.e., keyway) configurations as shown. The terms“tongue” and “key” are used interchangeably herein. Similarly, the terms“groove” and “keyway” are used interchangeably herein.

Specifically, the outer disk 1102 has a groove 1108 and the middle disk1104 has a tongue 1110 on a side facing the outer disk 1102 and thuscorresponds to and engages with the groove 1108. Similarly, the middledisk 1104 has a groove 1112 on a side that faces the outer disk 1106,and the outer disk 1106 has a tongue 1114 that corresponds to andengages with the groove 1112. The tongue 1110 is perpendicular to thegroove 1112. Accordingly, the middle disk 1104 is configured to slideradially with respect to the outer disks 1102 and 1106 as the disks1102, 1104, and 1106 rotate.

The unaligned input and output shafts are coupled to the outer disks1102 and 1106 and the middle disk 1104 transfers rotation of the inputshaft to the output shaft. Because the middle disk 1104 is configured toslide radially with respect to the outer disks 1102 and 1106, the effectof the misalignment between the input and output shafts is eliminated.

The coupling 1100 could be used to eliminate the eccentricity of thecycloid drives described above. As an example, referring back to FIG.2C, one of the outer disks 1102 and 1106 of the coupling 1100 may becoupled to the second ring 208. The output shaft may then be coupled tothe other outer disk. Thus, as the second ring 208 rotates in aneccentric manner, the ability of the middle disk 1104 to slide radiallywith respect to the outer disks 1102 and 1106 compensates for the effectof the eccentricity at the output shaft.

FIGS. 12A-12B illustrates a coupling configuration 1200 to compensatefor eccentricity at an output of a cycloid drive, in accordance with anexample implementation. The cycloid drive illustrated in FIGS. 12A-12Bis similar to the cycloid drive apparatus 216 with its first ring andsecond ring. The rollers and roller cage are omitted to reduce visualclutter in FIGS. 12A-12B.

In the configuration 1200, in contrast to the relatively large tongueand groove arrangement illustrated in FIG. 11, the cycloid drive has anarray of smaller tongues and grooves. By using multiple parallel smallertongues and grooves, the load capacity of the coupling increases for agiven volume, thus enabling the configuration 1200 to be more compactfor a give load capacity.

FIG. 12A illustrates an exploded view of the cycloid drive with oneviewing angle and FIG. 12B illustrates an exploded view from anotherviewing angle so that both sides of the components may be illustrated inthe Figures. The input shaft may be coupled to component 1201 at acenter of the component 1201 (i.e., the input shaft and the component1201 are concentric). A bearing 1202 is eccentrically mounted to theinput component 1201. The eccentrically mounted bearing 1202 is coupledto a second ring 1204 of the cycloid drive. The second ring 1204 couldbe disposed within a first ring 1206 (similar to the configuration ofthe second ring 208 and the first ring 200 in the apparatus 216).

Further, as shown in FIG. 12B, the second ring 1204 is configured tooperate as one of the output disks 1102 and 1106. The second ring 1204has a first side that faces toward the input shaft and a second sideopposite to the first side. The second side includes multiple tonguesand grooves 1208 as opposed to a single tongue (e.g., the tongue 1114)or a single groove (e.g., the groove 1108).

The configuration 1200 includes an intermediate disk 1210 that isequivalent to the middle disk 1104 in FIG. 11. Instead of the singletongue 1110 and the single groove 1112 of the disk 1104, the disk 1210has multiple tongues and grooves 1212 and 1214 on both sides of thedisk. Particularly, a first side of the disk 1210 faces toward thesecond ring 1204 and has the tongues and grooves 1212, whereas a secondside opposite to the first side has the tongues and grooves 1214. Thetongues and grooves 1212 are perpendicular to tongues and grooves 1214.

Disk 1216 is an output disk similar to either of the outer disks 1102and 1106. The disk 1216 has multiple tongues or grooves 1218 configuredto engage with the tongues or grooves 1214 of the disk 1210. Inoperation, as the second ring 1204 moves in an eccentric manner, thedisk 1210 slides radially with respect to both the second ring 1204 andthe output disk 1216, and thus the eccentricity of the output iseliminated.

FIGS. 13A-13B illustrate an exploded view of another configuration 1300to compensate for eccentricity at an output of a cycloid drive, inaccordance with an example implementation. The configuration 1300 useslinks to compensate for the eccentricity.

An input shaft may be coupled to an eccentrically mounted bearing 1302.The bearing 1302 is eccentric with respect to the input shaft asdescribed in previous configurations. The configuration 1300 includes asecond (internal) ring 1304 rotatable within a first ring 1306. Thesecond ring 1304 has a first side facing toward the input shaft and asecond side opposite the first side.

The configuration 1300 includes four links 1308A, 1308B, 1308C, and1308D, each link having two pegs. More or fewer links could be used, andeach link could have more or fewer pegs. The links 1308A, 1308B, 1308C,and 1308D are connected or coupled to each other and are disposed in aplane parallel to a respective plane of the second ring 1304. The pegsof the links 1308A, 1308B, 1308C, and 1308D protrude in a directionperpendicular to the plane.

The configuration 1300 also has an intermediate member 1310 having fourholes as shown. The configuration 1300 also has an output member 1312that has two holes, such as hole 1314. A subset the pegs of the links1308A, 1308B, 1308C, and 1308D face toward and are coupled to theintermediate member 1310 and the output member 1312, whereas anothersubset of pegs face toward and are coupled to the second ring 1304.

The dashed lines in FIG. 13B illustrate how half of the pegs of thelinks 1308A, 1308B, 1308C, and 1308D are coupled to holes in othermembers. For instance, a peg 1316 of the link 1308D extends beyond theintermediate member 1310 and is coupled to the hole 1314 of the outputmember 1312. A peg 1318 of the link 1308D is coupled to a hole 1319 ofthe intermediate member 1310. A peg 1320 of the link 1308B is coupled toa hole 1322 of the intermediate member 1310. A peg 1324 of the link1308B is coupled to a hole 1326 of the second ring 1304. Only half theconnections of the pegs with respective holes are shown to reduce visualclutter in the drawings.

Each peg is free to rotate within a corresponding hole the peg isreceived at or coupled thereto. As described in previous configurations,the second ring 1304 moves about in an eccentric manner. The lengths ofthe links 1308A, 1308B, 1308C, and 1308D should be substantially largerthan the amount of eccentricity of the second ring 1304 so as tocompensate for the eccentricity. The second ring 1304 causes, via thepegs coupled thereto, the links 1308A-D and the intermediate member 1310to move in a manner that eliminates or reduces the eccentric motion.Thus, the output member 1312 also moves about without the eccentricity.A shaft coupled to the output member would therefore rotate withouteccentric motion.

FIG. 14 illustrates another configuration 1400 to compensate foreccentricity of a cycloid drive, in accordance with an exampleimplementation. While the configurations shown previously include asingle second ring, the configuration 1400 includes a compound secondring 1401 having two second rings 1402 and 1404. The two second rings1402 and 1404 are affixed to each other, and thus rotate as one unit atthe same speed.

The configuration 1400 also include two corresponding first rings 1406and 1408, such that the second ring 1402 rotates within the first ring1406 and the second ring 1404 rotates within the first ring 1408. One ofthe first rings is grounded, i.e., fixed, while the other first ring isfree to rotate, i.e., floating.

Pitch diameters of the two second rings 1402 and 1404 differ by a smallor threshold amount. For example, the pitch diameter of the second ring1402 could be 55 mm and the pitch diameter of the second ring 1404 couldbe 50 mm. Similarly, the pitch diameters of the two first rings 1406 and1408 differ by a small or threshold amount, but the difference in pitchdiameters of the first rings 1406 and 1408 is equal to the difference inthe pitch diameters of the second rings 1402 and 1404. For example, thepitch diameter of the first ring 1406 could be 60 mm and the pitchdiameter of the first ring 1408 could be 55 mm.

FIG. 14 also shows an input shaft 1410 coupled to an eccentric component1412. The second rings 1402 and 1404 are mounted to a roller bearing1414 configured to encompass the eccentric component 1412. In operation,as the input shaft 1410 rotates, the second rings 1402 and 1404 bothmove in an eccentric manner within their respective first rings 1406 and1408, respectively.

If the first ring 1406 is considered as the “ground” ring (i.e., thefirst ring that does not move), then the output may be harvested fromthe first ring 1408, which is free to rotate or is floating. Thearrangement of the second ring 1404 and the first ring 1408 cancels theeccentricity of the rotation of the second ring 1402 within the firstring 1406. Thus, the output has no eccentricity. In this configuration,a large reduction ratio may be obtained as the reduction ratio of theconfiguration 1400 is the product of the two reduction ratios betweeneach first ring and second ring pair.

X. Conclusion

It should be understood that arrangements described herein are forpurposes of example only. As such, those skilled in the art willappreciate that other arrangements and other elements (e.g., machines,interfaces, orders, and groupings of operations, etc.) can be usedinstead, and some elements may be omitted altogether according to thedesired results.

While various aspects and implementations have been disclosed herein,other aspects and implementations will be apparent to those skilled inthe art. The various aspects and implementations disclosed herein arefor purposes of illustration and are not intended to be limiting, withthe true scope being indicated by the following claims, along with thefull scope of equivalents to which such claims are entitled. It is alsoto be understood that the terminology used herein is for the purpose ofdescribing particular implementations only, and is not intended to belimiting.

What is claimed is:
 1. An apparatus comprising: a first ring having anopen annular space and a series of variable-width cutouts disposed on aninterior peripheral surface of the first ring; a second ring rotatablewithin the open annular space of the first ring, wherein the second ringhas a respective series of variable-width cutouts disposed on anexterior peripheral surface of the second ring; and a plurality ofrollers disposed between, and configured to roll on, the interiorperipheral surface of the first ring and the exterior peripheral surfaceof the second ring and rotatable therebetween, wherein: the first ringhas a total number of variable-width cutouts and the second ring has atotal number of variable-width cutouts, with the total number ofvariable-width cutouts of the second ring being smaller than the totalnumber of variable-width cutouts of the first ring, and a total numberof the plurality of rollers being less than the total number ofvariable-width cutouts of the first ring and greater than the totalnumber of variable-width cutouts of the second ring.
 2. The apparatus ofclaim 1, wherein as a given roller of the plurality of rollers traversesa variable-width cutout of either the variable-width cutouts of thefirst ring or the variable-width cutouts of the second ring, a radialdistance between a center of the first ring and the given roller varies.3. The apparatus of claim 1, wherein at least one cutout of thevariable-width cutouts starts with a first width at a first end of thecutout, increases to a second width larger than the first width at acenter of the cutout, and narrows back to the first width at a secondend of the cutout.
 4. The apparatus of claim 1, further comprising: aroller cage disposed between the first ring and the second ring andconfigured to couple the plurality of rollers to each other, wherein theroller cage is rotatable in the open annular space of the first ring asthe plurality of rollers roll between the interior peripheral surface ofthe first ring and the exterior peripheral surface of the second ring.5. The apparatus of claim 4, wherein the roller cage couples theplurality of rollers such that the rollers are equidistant from eachother.
 6. The apparatus of claim 1, wherein the plurality of rollers arespherical.
 7. The apparatus of claim 1, wherein the plurality of rollerseach have a rhombic cross section in a plane parallel to an axis ofrotation of the second ring and a circular cross section in a planeperpendicular to the axis of rotation of the second ring.
 8. Theapparatus of claim 1, wherein the interior peripheral surface of thefirst ring has a variable-width groove disposed therein, wherein a widthof the variable-width groove varies between a first width and a secondwidth larger than the first width, wherein the variable-width cutoutscomprise regions of the variable-width groove that increase from thefirst width to the second width and back to the first width, and whereinthe variable-width cutouts are separated by portions of thevariable-width groove having the first width.
 9. The apparatus of claim1, wherein the first ring is fixed.
 10. An apparatus comprising: a firstring having an open annular space and a plurality of depressionsspatially arranged in series along an interior peripheral surface of thefirst ring, wherein a groove is disposed in the interior peripheralsurface including the plurality of depressions; a second ring rotatablewithin the open annular space of the first ring, wherein the second ringhas a respective plurality of depressions spatially arranged in seriesalong an exterior peripheral surface of the second ring, and wherein arespective groove is disposed in the exterior peripheral surfaceincluding the respective plurality of depressions; and a plurality ofrollers configured to engage, and roll within, the groove disposed inthe interior peripheral surface and the respective groove disposed inthe exterior peripheral surface as the second ring rotates within theopen annular space of the first ring.
 11. The apparatus of claim 10,wherein: the first ring has a total number of depressions and the secondring has a total number of depressions, with the total number ofdepressions of the second ring being smaller than the total number ofdepressions of the first ring, and a total number of the plurality ofrollers being less than the total number of depressions of the firstring and greater than the total number of depressions of the secondring.
 12. The apparatus of claim 10, further comprising: a roller cagedisposed between the first ring and the second ring and configured tocouple the plurality of rollers to each other, wherein the roller cageis rotatable in the open annular space as the plurality of rollers rollbetween the interior peripheral surface of the first ring and theexterior peripheral surface of the second ring.
 13. The apparatus ofclaim 10, wherein either the first ring or the second ring is fixed. 14.The apparatus of claim 10, wherein as a given roller of the plurality ofrollers traverses the groove and the plurality of depressions of thefirst ring and the respective groove and the respective plurality ofdepressions of the second ring, a radial distance between a center ofthe first ring and the given roller varies.
 15. An apparatus comprising:a first ring having an open annular space and a variable-width groovedisposed on an interior peripheral surface of the first ring, whereinthe variable-width groove of the first ring defines a plurality ofregions, such that at least one region of the plurality of regionsstarts with a first width at a first end of the region, increases to asecond width larger than the first width at a center of the region, andnarrows back to the first width at a second end of the region; a secondring rotatable within the open annular space of the first ring, whereinthe second ring has a respective variable-width groove disposed on anexterior peripheral surface of the second ring, wherein the respectivevariable-width groove of the second ring defines a respective pluralityof regions, such that at least one respective region of the respectiveplurality of regions starts with the first width at a respective firstend of the respective region, increases to the second width at arespective center of the respective region, and narrows back to thefirst width at a respective second end of the respective region; and aplurality of rollers disposed between, and configured to roll on, theinterior peripheral surface of the first ring and the exteriorperipheral surface of the second ring and rotatable therebetween whileengaging the variable-width groove of the first ring and the respectivevariable width groove of the second ring, wherein a total number ofregions defined by the variable-width groove of the second ring issmaller than a total number of regions defined by the respectivevariable-width groove of the first ring and a total number of theplurality of rollers is less than the total number of regions defined bythe variable-width groove of the first ring and greater than the totalnumber of regions defined by the respective variable-width groove of thesecond ring.
 16. The apparatus of claim 15, wherein as a given roller ofthe plurality of rollers traverses the variable-width groove of thefirst ring and the respective variable-width groove of the second ring,a radial distance between a center of the second ring and the givenroller varies.
 17. The apparatus of claim 15, wherein the variable-widthgroove is a first variable-width groove, wherein the respectivevariable-width groove is a first respective variable-width groove,wherein the first ring has a second variable-width groove disposedparallel to the first variable-width groove, and wherein the second ringhas a second respective variable-width grove disposed parallel to thefirst respective variable-width groove.
 18. The apparatus of claim 17,wherein at least one roller of the plurality of rollers comprises two ofside-by-side rollers coupled to each other and configured to traversethe first and second variable-width grooves of the first ring and thefirst and second respective variable-width grooves of the second ring.